English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

In this paper we obtain a result of the asymptotic behavior of the nth order equation r¿("' + f(t,u,u',... ,«("-1)) = 0 under some assumptions. For n = 2 and f{t,u,u') = f{t,u), it revises the result given by Jingcheng Tong, which is not true in general. Much work has been done on the asymptotic behavior of the second order equation (1) u" + fit,u)=0. Some results of it are based on the integral inequalities of Gronwall-Bihari type. Here we quote the Theorem B in [1] as a proposition, according to the author, which includes the Theorem in [2] as its special case. PROPOSITION. Let fit,u) be continuous on D: t > 0, -oo < u < oo. If there are two nonnegative and continuous functions vit), 0(t) for t > 0, and a continuous function giu) for u>0, such that (i) /j°° vit)<f>it) dt < co, (ii) for u > 0, giu) is positive and nondecreasing, (iii) |/(i, u)| < vit)(j>it)gi\u\/t) for t > 1, -oo < u < oo, then the equation (1) has solutions which are asymptotic to a + bt, where a,b are constants and 6^0. According to the proof it seems to be true that every solution w(i) of Equation (1) satisfies that u'(i) —► b £ 72 as t —► oo. But unfortunately, the Proposition does not hold in general because of two mistakes in its proof. The first error arises because Gic3)+f1 t;(s)0(s) ds may be outside the domain of G_1 (see (4) in [1]). Subsequently, the second error arises because one cannot hence conclude that f^° |/(s, u(s))| ds < oo. For example, the equation u" — (2/í4)u2 = 0 has a solution u = r2 which does not satisfy the property u'(£) —► b it —> oo) as depicted in the proof of the Proposition. For the same reason, the example in [1] cannot be true. In this paper we obtain a result of the asymptotic behavior of the nth order equation (2) u^A-fit,u,u',...,u^-^)=0. In a special case, for n = 2 and /(i,u,u') = fit,u), it revises the result in [1]. Received by the editors April 28, 1987 and, in revised form, July 8, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 34C99. ©1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page

Asymptotic behavior of a class of nonlinear differential equations of 𝑛th order